On the (co)girth of connected matroids ?

This article studies the girth and cogirth problems for connected matroids. The problem of finding the cogirth of a graphic matroid has been intensively studied, but studies on the equivalent problem for a linear matroid or a general matroid have been rarely reported. Based on the duality and connectivity of a matroid, we prove properties associated with the girth and cogirth of a matroid whose contraction or restriction is disconnected. Then, we devise algorithms that find the cogirth of a matroid M from the matroids associated with the direct sum components of the restriction of M . As a result, the problem of finding the (co)girth of a matroid can be decomposed into a set of smaller sub-problems, which helps alleviate the computation. Finally, we implement and demonstrate the application of our algorithms to linear matroids.